Multi-stage sodium heat engine for electricity and heat production

ABSTRACT

A multi-stage sodium heat engine is provided to convert thermal energy to electrical energy, the multi-stage sodium heat engine including at least a first stage, a second stage, and an electrical circuit operatively connecting the first stage and the second stage with an electrical load. One or more methods of powering an electrical load using a multi-stage sodium heat engine are also described.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication No. 62/222,064, titled: DUAL STAGE SODIUM HEAT ENGINE FORELECTRICITY AND HEAT PRODUCTION, filed Sep. 22, 2015, the entiredisclosure of which is hereby incorporated by reference.

FIELD

One or more embodiments of the one or more present inventions relate, ingeneral, to sodium heat engines to produce electricity from low costheat sources, including solar energy, waste heat, and other forms ofheat. More particularly, one or more embodiments described hereinpertain to a multi-stage sodium heat engine, such as a dual-stage sodiumheat engine.

BACKGROUND

The Alkali Metal Thermal to Electric Converter (AMTEC) was developedapproximately 50 years ago to directly convert thermal energy toelectrical energy. A typical AMTEC system 10 is illustrated in FIG. 1.In a typical AMTEC system 10, an alkali metal, such as sodium 12, istransferred around a closed thermodynamic cycle. A porous anode 14generates sodium-ions 16 which transport across a solid electrolyte 18,and electrons which travel from the anode 14 through an external circuit20 to perform electrical work on load 22 to the low-pressure porouscathode 24, where they recombine with the sodium ions 16. In a typicalAMTEC cycle, the sodium ion conduction between a high-pressure (orhigh-activity) region and a low-pressure (or low-activity) regionthrough the sodium-ion conductive solid electrolyte 18 isthermodynamically nearly equivalent to an isothermal expansion of sodiumvapor between the same high and low pressures.

As shown in FIG. 1, the AMTEC system 10 further includes an evaporator26, typically at a temperature greater than about 1000K, to vaporizeliquid sodium 30 to produce sodium vapor 12 and a condenser 28,typically at a temperature less than about 700K, to reduce temperatureand reform the liquid sodium. The liquid sodium 30 is then returned tothe evaporator 26 by an EM pump or passive wick 32.

In a typical AMTEC system 10, the sodium ion conductive solidelectrolyte 18, which is a conductor of positive ions but an insulatorto electrons, may be a β″-alumina solid electrolyte (BASE). At the sideof the solid electrolyte 18 closer to the anode 14, heat is added tovaporize the sodium 12, resulting in a temperature greater than about1000K and a pressure of over 20 kPa. At the side of the solidelectrolyte closer to cathode 24, heat is removed to cool the sodium 12to a temperature below 700K and a pressure less than 100 Pa. Despitebeing the “cold” side, the temperature is maintained at a relativelyhigh temperature to maintain the sodium in liquid form.

At the anode 14 surface of the sodium ion conductive solid electrolyte18, the sodium atoms 12 in the vapor are oxidized, which releaseselectrons to flow through external circuit 20, powering load 22. Theresulting sodium ions 16 absorb the latent heat of vaporization. Due tothe high pressure difference across the sodium ion conductive solidelectrolyte 18 and its differential conductivity between electrons andsodium ions 16, the sodium ions 16 are transferred through the sodiumion conductive solid electrolyte 18 to the cathode 24, while theelectrodes provide a conduction path for the free electrons to transferacross the external load 20 doing useful work on their way to thecathode 24 where they are recombined with the sodium ions 16 to formsodium vapor 12. At the cathode 24 side of the sodium ion conductivesolid electrolyte 18, the sodium vapor 12 releases its latent heat ofvaporization at the condenser 28 to form liquid (molten) sodium 30 whichis transported back to the evaporator 26 by an electromagnetic pump orpassive wick mechanism 32 to restart the cycle.

Typical multi-membrane β″-alumina AMTEC systems, which include parallelconnection of multiple membranes in a system 10, may be operated attemperatures between about 1300 K on the hot side and 700 K on the coolside, with typical open circuit voltages of 0.7 V to 1.4 V for a singlecell. While the Carnot engine efficiency limit for a 1300 K to 700 Koperation is about 0.46, measured efficiencies have been reported in therange of 0.2-0.4 by Wu, S., et al. “A parabolic dish/AMTEC solar thermalpower system and its performance evaluation,” Applied Energy, 2010, 87,452. Without wishing to be held to any particular theory, lossesefficiency for typical AMTECs are believed to be related to Na leak dueto compromised seal, ohmic contact losses, and electronic conduction inβ″-alumina.

While β″-alumina phase transformation has been predicted and observed attemperatures above 1273 K, thermal degradation of β″-alumina above 1100K has been attributed to reaction with vapor phase of Na and thermalbreakdown at high temperatures.

Improvements in the foregoing are desired.

SUMMARY

The present disclosure provides a multi-stage sodium heat engine andmethods of using the same.

It is to be understood that the one or more present inventions disclosedherein may include a variety of different versions or embodiments, andthis Summary is not meant to be limiting or all-inclusive. This Summaryprovides some general descriptions of some of the embodiments, but mayalso include some more specific descriptions of other embodiments.

In one exemplary embodiment, a multi-stage sodium heat engine isprovided. The sodium heat engine includes a first stage comprising afirst porous anode and a first porous cathode separated by a firstsodium conductive solid electrolyte, the first stage configured toprovide expansion of a first stream of vaporized sodium from a pressureP₄ to a pressure P₃, where P₄ is greater than P₃, by ionizing thevaporized sodium to form sodium ions, the sodium ions being transferablethrough the first sodium conductive solid electrolyte, the first porouscathode reducing the sodium ions to form a second stream of vaporizedsodium. The sodium heat engine further includes a second stagecomprising a second porous anode and a second porous cathode separatedby a second sodium conductive solid electrolyte. The second stage isoperably connected to the first stage and configured to provideexpansion of the second stream of vaporized sodium from a pressure P₂ toa pressure P₁, where P₂ is greater than P₁, by ionizing the vaporizedsodium to form sodium ions, the sodium ions being transferrable throughthe second sodium conductive solid electrolyte, the second porouscathode reducing the sodium ions to form a third stream of vaporizedsodium. The sodium heat engine further includes an electrical circuitoperatively connecting the first porous anode and the second porouscathode with an electrical load.

In one more particular embodiment, the first sodium conductive solidelectrolyte comprises a β″-alumina electrolyte, and more particularlycomprises a sodium ion conductive solid electrolyte selected from thegroup consisting of: (Na₂O)_(1-x)11Al₂O₃ where x is from 0 to 0.75, moreparticularly from 0.10 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33 Al₂O₃. In a more particular embodiment, thesodium ion conductive solid electrolyte 12 has the formula(Na₅LiAl₃₂O₅₁), and even more particularly, comprises a sodium ionconductive solid electrolyte of the formula Na₅LiAl₃₂O₅₁.

In one more particular embodiment of any of the above embodiments, thesecond sodium conductive solid electrolyte comprises a material selectedfrom the group consisting of: Na₃Zr₂Si₂PO₁₂; Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂where x is from 1.6 to 2.4; Y doped NaSICON of the formula(Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂,Na_(1+x)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12;Fe-doped NaSICON of the formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂,wherein Re is selected from the group consisting of Y, Gd, Dy, and Nd;and Na₄ZrSi₃O₁₂, and even more particularly comprises a sodium ionconductive solid electrolyte of the formula Na₄ZrSi₃O₁₂. In another moreparticular embodiment, the second sodium conductive solid electrolytecomprises a β″-alumina electrolyte, and more particularly, a materialselected from the group consisting of: (Na₂O)_(1−x)11Al₂O₃ where x isfrom 0.1 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃.

In one more particular embodiment of any of the above embodiments, thesodium heat engine further includes a heat source for providing aquantity of heat to the second stream of vaporized sodium between thefirst stage and the second stage.

In one more particular embodiment of any of the above embodiments, thesodium heat engine further includes a third stage comprising a thirdporous anode and a third porous cathode separated by a third sodiumconductive solid electrolyte, the third stage operably connected to thesecond stage and configured to provide expansion of the third stream ofvaporized sodium by ionizing the vaporized sodium to form sodium ions,the sodium ions being transferrable through the third sodium conductivesolid electrolyte.

In one more particular embodiment of any of the above embodiments, thepressure P₄ is from about 160,000 Pa to about 10,000 Pa. In another moreparticular embodiment of any of the above embodiments, the first stageis configured to provide isothermal expansion of the vaporized sodium ata temperature of about 1100 K to about 1250 K. In another moreparticular embodiment of any of the above embodiments, the second stageis configured to provide isothermal expansion of the vaporized sodium ata temperature from about 850 K to about 400 K.

In one more particular embodiment of any of the above embodiments, thesodium heat engine further includes a vaporizer to vaporize a flow ofmolten sodium to form the first stream of vaporized sodium at thepressure P₄. In a more particular embodiment, the sodium heat enginefurther includes a heat source for providing a quantity of heat to thevaporizer, such as a concentrating solar power system. In another moreparticular embodiment, the sodium heat engine further includes acondenser to condense the third stream of vaporized sodium to form amolten stream of sodium. In an even more particular embodiment, thesodium heat engine further includes a pump or passive wick positionedbetween the condenser and the vaporizer, wherein the pump or passivewick allows for flow of molten sodium between the condenser and thevaporizer.

In one more particular embodiment of any of the above embodiments, thesodium heat engine has a Carnot efficiency of at least 0.5. In oneanother particular embodiment of any of the above embodiments, thesodium heat engine has an open-circuit electrochemical potential of atleast 0.55 V.

In one exemplary embodiment, a method of powering an electrical load isprovided. The method includes providing a first stream of sodium vapor;oxidizing the first stream of sodium vapor with a first anode to producesodium ions and electrons and transporting the sodium ions across afirst sodium conductive solid electrolyte; reducing the sodium ions witha first cathode to form a second stream of sodium vapor; oxidizing thesecond stream of sodium vapor with a second anode to produce sodium ionsand electrons and transporting the sodium ions across a second sodiumconductive solid electrolyte; and reducing the sodium ions with a secondcathode to form a third stream of sodium vapor. The electrical load ispart of an electronic circuit operatively connecting the first anode andthe second cathode.

In one more particular embodiment, the first sodium conductive solidelectrolyte comprises a β″-alumina electrolyte, and more particularlycomprises a sodium ion conductive solid electrolyte selected from thegroup consisting of: (Na₂O)_(1−x)11Al₂O₃ where x is from 0 to 0.75, moreparticularly from 0.10 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃. In a more particular embodiment, thesodium ion conductive solid electrolyte 12 has the formula(Na₅LiAl₃₂O₅₁), and even more particularly, comprises a sodium ionconductive solid electrolyte of the formula Na₅LiAl₃₂O₅₁.

In another more particular embodiment, the second sodium conductivesolid electrolyte comprises a material selected from the groupconsisting of: Na₃Zr₂Si₂PO₁₂; Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂ where x isfrom 1.6 to 2.4; Y doped NaSICON of the formula(Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂,Na_(1+x)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12;Fe-doped NaSICON of the formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂,wherein Re is selected from the group consisting of Y, Gd, Dy, and Nd;and Na₄ZrSi₃O₁₂, and even more particularly comprises a sodium ionconductive solid electrolyte of the formula Na₄ZrSi₃O₁₂. In another moreparticular embodiment, the second sodium conductive solid electrolytecomprises a β″-alumina electrolyte, and more particularly, a materialselected from the group consisting of: (Na₂O)_(1-x)11Al₂O₃ where x isfrom 0.1 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃.

In one exemplary embodiment, a multi-stage sodium heat engine isprovided. The multi-stage sodium heat engine includes a first stagecomprising a first sodium conductive solid electrolyte, the first stageconfigured to provide expansion of a first stream of vaporized sodium byionizing the vaporized sodium to form sodium ions, the sodium ions beingtransferable through the first sodium conductive solid electrolyte. Inone more particular embodiment, the first sodium conductive solidelectrolyte comprises a β″-alumina electrolyte, and more particularlycomprises a sodium ion conductive solid electrolyte selected from thegroup consisting of: (Na₂O)_(1−x)11Al₂O₃ where x is from 0 to 0.75, moreparticularly from 0.10 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃. In a more particular embodiment, thesodium ion conductive solid electrolyte has the formula (Na₅LiAl₃₂O₅₁),and even more particularly, comprises a sodium ion conductive solidelectrolyte of the formula Na₅LiAl₃₂O₅₁.

The multi-stage sodium heat engine also includes a second stagecomprising a second sodium conductive solid electrolyte, the secondstage configured to provide expansion of a second stream of vaporizedsodium by ionizing the vaporized sodium to form sodium ions, the sodiumions being transferable through the second sodium conductive solidelectrolyte. In one more particular embodiment, the second sodiumconductive solid electrolyte comprises a material selected from thegroup consisting of: Na₃Zr₂Si₂PO₁₂; Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂ where xis from 1.6 to 2.4; Y doped NaSICON of the formula(Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂, Na_(1+x)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12; Fe-doped NaSICON ofthe formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂, wherein Re isselected from the group consisting of Y, Gd, Dy, and Nd; andNa₄ZrSi₃O₁₂, and even more particularly comprises a sodium ionconductive solid electrolyte of the formula Na₄ZrSi₃O₁₂. In another moreparticular embodiment, the second sodium conductive solid electrolytecomprises a β″-alumina electrolyte, and more particularly, a materialselected from the group consisting of: (Na₂O)_(1−x)11Al₂O₃ where x isfrom 0.1 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃.

The multi-stage sodium heat engine further includes an electricalcircuit operatively connecting the first stage and the second stage withan electrical load, wherein a Carnot efficiency of the multi-stagesodium heat engine is at least 0.5.

In a more particular embodiment of any of the above embodiments, thesecond stage receives the second stream of vaporized sodium from thefirst stage. In another more particular embodiment, the multi-stagesodium heat engine further includes a third stage comprising a thirdsodium conductive solid electrolyte, the third stage configured toprovide expansion of a third stream of vaporized sodium by ionizing thevaporized sodium to form sodium ions, the sodium ions being transferablethrough the third sodium conductive solid electrolyte, wherein theelectrical circuit operatively connects the first stage, the secondstage, and the third stage with the electrical load.

Various components are referred to herein as “operably associated.” Asused herein, “operably associated” refers to components that are linkedtogether in operable fashion, and encompasses embodiments in whichcomponents are linked directly, as well as embodiments in whichadditional components are placed between the two linked components.

BRIEF DESCRIPTION OF THE DRAWINGS

The above-mentioned and other features and advantages of thisdisclosure, and the manner of attaining them, will become more apparentand the invention itself will be better understood by reference to thefollowing description of embodiments of the invention taken inconjunction with the accompanying drawings, wherein:

FIG. 1 illustrates a typical AMTEC system known in the prior art;

FIG. 2 illustrates an exemplary multi-stage sodium heat engine includinga first stage and a second stage; and

FIG. 3 illustrates an exemplary multi-stage sodium heat engine includingthree stages.

Corresponding reference characters indicate corresponding partsthroughout the several views. The exemplifications set out hereinillustrate exemplary embodiments of the invention and suchexemplifications are not to be construed as limiting the scope of theinvention in any manner.

DETAILED DESCRIPTION

The present disclosure provides a multi-stage sodium heat engine andmethods related thereto. For purpose of this disclosure, the descriptiondescribes a dual-stage sodium heat engine; however, as described inadditional detail below, it is to be understood that additional stagesare encompassed by this disclosure.

An exemplary dual-stage sodium heat engine 110 is illustrated in FIG. 2.The dual-stage sodium heat engine 110 includes a first stage 112 and asecond stage 114. The first stage 112 illustratively operates at arelatively higher temperature range, while the second stage 114illustratively operates at a relatively lower temperature range. Withoutwishing to be held to any particular theory, it is believed that the useof multiple stages allows for a more efficient utilization of heataround a wider temperature and up to 400° C. for conversion toelectricity, particularly at lower temperatures in the second stage 114.In some more particular embodiments, the use of different first stagesolid electrolytes 124 and second stage solid electrolytes 136 allow forselection of materials based on the different operating temperatures inthe first stage 112 and second stage 114.

As shown in FIG. 2, a first quantity of heat Q₁ is supplied from a heatsource 116, such as a concentrating solar power (CSP) system, to theevaporator 118, vaporizing liquid sodium 120 at temperature T₁ andpressure P₁ to form sodium vapor 122 at increased temperature T₃ andpressure P₃, where the pressure P₃ is the sodium vapor pressure at thegiven temperature T₃.

Referring to FIG. 2, the first stage 112 illustratively includes asodium ion conductive solid electrolyte 124, a porous anode 126, and aporous cathode 128. Sodium vapor 122 at temperature T₄ and pressure P₄enters the first stage 112. The first stage 112 includes the expansionof sodium 122 from pressure P₄ to pressure P₃ (P₄>P₃) from the transferof sodium ions across the sodium ion conductive solid electrolyte 124,such as β″-alumina electrolyte, at high temperature T₄. At porouscathode 128, the sodium ions are rejoined to electrons from externalcircuit 130 to form sodium vapor 132 at temperature T₃ and pressure P₃.

In some exemplary embodiments, T₄ is about 1100 K to about 1250 K, P₄ isabout 10,000 Pa to about 160,000 Pa, or more particularly P₄ is about60,000 Pa to about 160,000 Pa, T₃ is from about 700K to about 850K, andP₃ is about 110 Pa to about 9000 Pa, or more particularly P₃ is about110 Pa to about 2300 Pa. In a more particular embodiment, T₄ is about1200 K and P₄ is about 150,000 Pa, and T₃ is about 850 K and P₃ is about2300 Pa.

Exemplary first stage materials for sodium ion conductive solidelectrolyte 124 include Beta Alumina (β-alumina and β″-alumina)electrolyte BASE materials having the formula (Na₂O)_(1−x)11Al₂O₃, wherex is from 0 to 0.75, or more particularly from 0.10 to 0.75;Na_(1.72)Al_(10.66)Li_(0.30)O₁₇ (Jorgensen et al., 1981); orNa_(1.67)(MgO)_(1.33)10.33Al₂O₃ (Bourke et al., 1980). In a moreparticular embodiment, the sodium ion conductive solid electrolyte 12has the formula (Na₅LiAl₃₂O₅₁). Suitable materials for the first stagesodium ion conductive solid electrolyte 124 include β-aluminaelectrolyte materials available from Ceramatec, Inc.

As shown in FIG. 2, a second quantity of heat Q₂ may optionally be addedto the sodium vapor 132 between the first stage 112 and second stage114. Referring to FIG. 2, the second quantity of heat Q₂ isillustratively provided by a heat source 134. In a more particularembodiment, the heat source 134 is also heat source 116, such as aconcentrating solar power (CSP) system. The addition of the secondquantity of heat Q₂ increases the temperature of the sodium vapor 132from T₃ to T₂ and the pressure from P₃ to P₂. In other exemplaryembodiments, no heat Q₂ is added and T₃=T₂ and P₃=P_(2.) In someexemplary embodiments, increasing the temperature and pressure of thesodium vapor 132 prior to the second stage 114 enhances the performanceof the second stage 114.

Referring again to FIG. 2, the second stage 114 illustratively includesa sodium ion conductive solid electrolyte 136, a porous anode 138, and aporous cathode 140. Sodium vapor 132 at temperature T₂ and pressure P₂enters the second stage 14. The second stage 114 includes the expansionof sodium 132 from pressure P₂ to pressure P₁(P₂>P₁) from the transferof sodium ions across the sodium ion conductive solid electrolyte 136,at temperature T₂. At porous cathode 140, the sodium ions are rejoinedto electrons from external circuit 142 to form sodium 144 at temperatureT₁ and pressure P₁.

In some exemplary embodiments, T₂ is about 700 K to about 850 K, P₂ isabout 110 Pa to about 2330 Pa, T₁ is about 400 K to about 600 K, and P₁is about 0.00020 Pa to about 5.7 Pa.

Exemplary second stage electrolyte materials for sodium ion conductivesolid electrolyte 136 include Beta Alumina electrolyte BASE materials asdescribed with respect to sodium ion conductive solid electrolyte 124above, and NaSICON (Na Super Ion CONducting) sodium ion conductive solidelectrolyte materials. NaSICON operation in molten sodium has beensuccessfully demonstrated at 453 K for 1,000 hours at a current densityof 100 mA/cm² for battery applications. Additionally, NaSICON hasexhibited stability in molten sodium at 523 K that provides higherefficiency, cost savings, and wider temperature range for operationcompared to conventional Alkali Metal Thermal to Elect Converters(AMTEC).

The NaSICON sodium ion conductive solid electrolyte material maycomprise any known or novel NaSICON-type material that is suitable foruse with the described sodium heat engine. Exemplary NaSICON materialsare described in U.S. Pat. No. 8,246,863, the disclosures of which arehereby incorporated by reference in their entirety. Exemplary suitableexamples of NaSICON-type compositions include, but are not limited to,Na₃Zr₂Si₂PO₁₂, Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂ (where x is selected from 1.6to 2.4), Y-doped NaSICON (Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂,Na_(1+x)Zr_(2−y)Y_(y) Si_(x)P_(3−x)O_(12−y) (where x=2, y=0.12),Fe-doped NaSICON (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂Na₅ReSi₄O₁₂ (where Re isselected from Y, Gd, Dy, and Nd), and Na₄ZrSi₃O₁₂. Suitable materialsfor the second stage sodium ion conductive solid electrolyte 136NaSelect™ solid-electrolyte materials available from Ceramatec, Inc.

As mentioned above, although the exemplary dual-stage sodium heat engine110 in FIG. 2 is illustrated as comprising a first stage 112 and asecond stage 114, in other embodiments, a multi-stage sodium heat enginemay include more than two stages. For example, FIG. 3 illustrates anexemplary dual stage sodium heat engine 110′ including a first stage112, a second stage 114, and a third stage 154. In other embodiments(not shown), a multi-stage sodium heat engine may include additionalstages, such as a fourth stage, a fifth stage, a sixth stage, etc., in amanner similar to the inclusion of a third stage 154 to the exemplarysodium heat engine 110′ illustrated in FIG. 3.

As illustrated in FIG. 3, an optional fourth quantity of heat Q₄ isillustratively provided by a heat source 156. In a more particularembodiment, the heat source 156 is also heat source 116 and/or heatsource 134, such as a concentrating solar power (CSP) system. Theaddition of the second quantity of heat Q₂ increases the temperature ofthe sodium vapor exiting second stage 114 from T₅ to T₆ and the pressureof the sodium vapor exiting second stage 114 from P₅ to P₆. The thirdstage 154 illustratively includes a sodium ion conductive solidelectrolyte 158, a porous anode 160, and a porous cathode 162. Thirdstage 154 illustratively includes a solid sodium ion conductive solidelectrolyte material 158, which may be formed from the same or differentmaterial than solid electrolyte material 124 and/or solid electrolytematerial 136. Sodium vapor exiting the second stage 136 enters the third154 stage. The third stage 154 includes the isothermal expansion ofsodium vapor from to pressure P₁ by sodium ion transfer through thesodium ion conductive solid electrolyte 158, while electrons are removedby external circuit 164 to porous anode 140 of the second stage 114. Atporous cathode 162, the sodium ions are rejoined to electrons fromexternal circuit 142 to form sodium 144 at temperature T₁ and pressureP₁.

Referring again to FIG. 2, the sodium 144 is condensed to liquid sodium148 by condenser 150, releasing the quantity of heat Q₃. Heat removal(Q₃) by a condenser occurs at a low temperature T₁, where T₁ issufficient to cause condensation of sodium vapor to molten sodium. Insome embodiments, the heat removal (Q₃) provides thermal energy forpreheating transfer fluid for the concentrating solar power (CSP) systemor for low temperature industrial applications. In one embodiment, thetemperature T₁ is as low as about 370 K, about 400 K, about 425 K, ashigh as about 450 K, about 500 K, about 525 K, or within any rangedefined between any two of the foregoing values, such as between about370 K and about 525 K, or between about 400 K to about 450 K, forexample.

The liquid sodium 148 is illustratively transported back to theevaporator 118 by a pump or passive wick mechanism 152.

Referring again to FIG. 2, electrical load 146 is illustratively poweredbetween the porous anode 126 of the high pressure chamber of first stage112 and the cathode of the low pressure chamber of the second stage 114.The porous cathode 128 of first stage 112 and the anode of second stage114 electrically connected to complete the circuit 130, 142 throughelectrical load 146.

The β″-alumina electrolyte materials for sodium ion conductive solidelectrolyte 124 of the first stage 112 is suitable for typical operationat temperatures from about 1150 K and about 800 K. The NaSICON materialsfor sodium ion conductive solid electrolyte 136 of the second stage 114is suitable for operation at typical temperatures of about 800 K andlower. The β″-alumina materials for sodium ion conductive solidelectrolyte 136 of the second stage 114 is also suitable for operationat typical temperatures of about 800 K and lower. Thus, a multi-stagesodium heat engine, including a dual-stage sodium heat engine, is ableto operate at a broader temperature range compared to conventional AMTECdevices. In addition, the first stage 112 of a multi-stage sodium heatengine, including a dual-stage sodium heat engine 110, is able tooperate at an upper temperature that is sufficiently low to maintain thestability and useful life of the β″-alumina solid electrolyte.

In some exemplary embodiments, the dual-stage sodium heat engine 110provides improved overall efficiency of CSP-based hybrid solar energyconverters and inexpensively generates both heat and electricity.Suitable materials for fabricating the dual-stage sodium heat engineinclude ceramic coated stainless steel or alumina and other suitablematerials thermally and chemically resistant to molten and vapor phasesodium. In some embodiments, a high temperature glass seal between thesolid electrolytes and various components of the dual-stage sodium heatengine 110 is provided to enable leak tight operation of the dual-stagesodium heat engine 110.

Theoretical Carnot Efficiency

Theoretical calculations show that a multi-stage sodium heat engine canprovide a higher theoretical efficiency compared to a typical AMTECsingle-stage engine.

Table 1 shows the Carnot efficiency (η) and the open-circuitelectrochemical potential (V^(OC)) for the dual-stage sodium heat engineat various T₁ values and a constant T₂ value of 1153 K as given byNernst equation, V^(OC)=R*(T₂−T₁)*ln(P₂/P₁)/(n*F). Here, n is the numberof electrons (1) in Na oxidation and reduction reactions, R is theuniversal gas constant (8.314 J·mol⁻¹·K⁻¹), and F is the Faraday'sconstant (96,485 C·mol⁻¹).

In the disclosed heat engine, heat is supplied by solar concentration,and sodium is the working fluid to drive the electrochemical unit thatgenerates electricity.

For the determination of Carnot efficiencies below, a dual-stage sodiumheat engine was modeled as shown in FIG. 2. The overall efficiency wasdetermined by combining the efficiencies determined for each stageseparately. The first stage solid electrolyte 124 was β″-alumina with anisothermal expansion temperature T₃ of 1153 K. The second stage solidelectrolyte 136 was NaSICON with an isothermal expansion temperature T₁as indicated in Table 1. The corresponding sodium vapor pressures P₃ fora T₃ of 1153 is 100,000 Pa, and the sodium vapor pressure P₁ for eachtemperature T₁ is shown in Table 1. In some exemplary embodiments, theNaSICON temperature T₁ provides a sufficiently high vapor pressure forsodium mass transfer, as well as a high thermal driving force for alarge Nernst potential.

TABLE 1 V^(OC) and η values at various T₁ temperatures T₁ (K) P₁ (Pa)V^(OC) (V) η 371 1.65*10⁻⁵ 1.36 0.68 421 8.67*10⁻⁴ 1.02 0.63 4711.97*10⁻² 0.77 0.59 523 2.69*10⁻¹ 0.57 0.55

As shown in Table 1, a dual-stage sodium heat engine 110 as illustratedin FIG. 2 can provide a theoretical efficiency of about 0.68, comparedto a reported theoretical efficiency of only 0.46 for an AMTECsingle-stage engine. The ideal thermodynamic cycle efficiency is >95% ofCarnot with thermal, ohmic and other resistances at operatingtemperature a practical efficiency of around 80% of the ideal cycleefficiency at an overall efficiency of >40% can be realized.

Without wishing to be held to any particular theory, it is believed thatthe use of different solid electrolytes in the first and second stages,such as the use of β″-alumina in the first stage and NaSICON in thesecond stage, provides a large driving force in the form of the greatertemperature difference enabling high open-circuit electrochemicalpotential. Low temperature NaSICON operation allows reduction in hightemperature T₃ to limit β″-alumina thermal degradation, yet maintainhigh thermal driving force and high Carnot efficiency. Further, NaSICONprovides higher conductivity than β″-alumina conductivity at lowertemperatures. In some exemplary embodiments, the dual-stage sodium heatengine will achieve current density of 100-200 mA/cm² at 1 V andthermal-to-electric conversion efficiency of 0.45.

In one embodiment, the Carnot efficiency for the dual-stage sodium heatengine is as little as 0.5, 0.55, 0.6, as high as 0.63, 0.65, 0.68, 0.7or within any range defined between any two of the foregoing values,such as between 0.5 and 0.7, 0.55 and 0.68, or 0.6 and 0.68, forexample. In one embodiment, the V^(OC) for the dual-stage sodium heatengine is as little as 0.55 V, 0.57 V, 0.6 V, 0.75 V, as high as 1.0 V,1.1 V, 1.2 V, 1.3 V, 1.4 V, 1.5 V, or within any range defined betweenany two of the foregoing values, such as between 0.5 V and 1.5 V,between 0.55 and 1.4 V, or between 0.57 and 1.36 V, for example.

While this invention has been described as having exemplary designs, thepresent invention can be further modified within the spirit and scope ofthis disclosure. This application is therefore intended to cover anyvariations, uses, or adaptations of the invention using its generalprinciples. Further, this application is intended to cover suchdepartures from the present disclosure as come within known or customarypractice in the art to which this invention pertains and which fallwithin the limits of the appended claims.

What is claimed is:
 1. A sodium heat engine comprising: a first stagecomprising first high pressure chamber which comprises a first porousanode and a first porous cathode separated by a first sodium conductivesolid electrolyte, the first stage configured to provide expansion of afirst stream of vaporized sodium from a pressure P₄ to a pressure P₃,where P₄ is greater than P₃, by ionizing the vaporized sodium to formsodium ions, the sodium ions being transferable through the first sodiumconductive solid electrolyte, the first porous cathode reducing thesodium ions to form a second stream of vaporized sodium; a second stagecomprising a second low pressure chamber which comprises a second porousanode and a second porous cathode separated by a second sodiumconductive solid electrolyte, the second stage operably connected to thefirst stage, and configured to provide expansion of the second stream ofvaporized sodium from a pressure P₂ to a pressure P₁, where P₂ isgreater than P₁, by ionizing the vaporized sodium to form sodium ions,the sodium ions being transferable through the second sodium conductivesolid electrolyte, the second porous cathode reducing the sodium ions toform a third stream of vaporized sodium; and an electrical circuitoperatively connecting the first porous anode and the second porouscathode with an electrical load wherein the first sodium conductivesolid electrolyte comprises a β″-alumina electrolyte.
 2. The sodium heatengine of claim 1, wherein the β″-alumina electrolyte is selected fromthe group consisting of: (Na₂O)_(1−x)11Al₂O₃ where x is from 0.1 to0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃.
 3. The sodium heat engine of claim 1,wherein the β″-alumina electrolyte comprises a sodium ion conductivesolid electrolyte of the formula Na₅LiAl₃₂O₅₁.
 4. The sodium heat engineof claim 1, wherein the pressure P₄ is from about 160,000 Pa to about10,000 Pa.
 5. The sodium heat engine of claim 1, wherein the first stageis configured to provide expansion of the sodium at a temperature ofabout 1100 K to about 1250 K.
 6. The sodium heat engine of claim 1,wherein the second sodium conductive solid electrolyte comprises amaterial selected from the group consisting of: Na₃Zr₂Si₂PO₁₂;Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂ where x is from 1.6 to 2.4; Y doped NaSICONof the formula (Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂,Na_(1+x)Zr_(2−y)Y Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12;Fe-doped NaSICON of the formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂,wherein Re is selected from the group consisting of Y, Gd, Dy, and Nd;and Na₄ZrSi₃O₁₂.
 7. The sodium heat engine of claim 6, wherein thesecond sodium conductive solid electrolyte comprises Na₃Zr₂Si₂PO₁₂. 8.The sodium heat engine of claim 6, wherein the second sodium conductivesolid electrolyte further comprises a β″-alumina electrolyte.
 9. Thesodium heat engine of claim 8, wherein the second sodium conductivesolid electrolyte comprises a material selected from the groupconsisting of: (Na₂O)_(1−x)11Al₂O₃ where x is from 0.1 to 0.75;Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; and Na_(1.67)(MgO)_(1.33)10.33Al₂O₃.10. The sodium heat engine of claim 1, wherein the second stage isconfigured to provide expansion of the sodium at a temperature fromabout 850 K to about 400 K.
 11. The sodium heat engine of claim 1,further comprising a vaporizer to vaporize a flow of molten sodium toform the first stream of vaporized sodium at the pressure P₄.
 12. Thesodium heat engine of claim 11, further comprising a heat source forproviding a quantity of heat to the vaporizer.
 13. The sodium heatengine of claim 12, wherein the heat source is a concentrating solarpower system.
 14. The sodium heat engine of claim 12, further comprisinga condenser to condense the third stream of vaporized sodium to form amolten stream of sodium.
 15. The sodium heat engine of claim 14, furthercomprising a pump or passive wick positioned between the condenser andthe vaporizer, wherein the pump or passive wick allows for flow ofmolten sodium between the condenser and the vaporizer.
 16. The sodiumheat engine of claim 1, further comprising a second heat source forproviding a quantity of heat to the second stream of vaporized sodiumbetween the first stage and the second stage.
 17. The sodium heat engineof claim 1, wherein the Carnot efficiency of the sodium heat engine isat least 0.5.
 18. The sodium heat engine of claim 1, wherein theopen-circuit electrochemical potential for the sodium heat engine is atleast 0.55 V.
 19. The sodium heat engine of claim 1, further comprisinga third stage comprising a third porous anode and a third porous cathodeseparated by a third sodium conductive solid electrolyte, the thirdstage operably connected to the second stage and configured to provideexpansion of the third stream of vaporized sodium by ionizing thevaporized sodium to form sodium ions, the sodium ions being transferablethrough the third sodium conductive solid electrolyte.
 20. A method ofpowering an electrical load using the sodium heat engine of claim 1, themethod comprising: providing a first stream of sodium vapor; oxidizingthe first stream of sodium vapor with the first porous anode to producesodium ions and electrons and transporting the sodium ions across afirst sodium conductive solid electrolyte; reducing the sodium ions withthe first porous cathode to form a second stream of sodium vapor;oxidizing the second stream of sodium vapor with the second porous anodeto produce sodium ions and electrons and transporting the sodium ionsacross a second sodium conductive solid electrolyte; and reducing thesodium ions with the second porous cathode to form a third stream ofsodium vapor; wherein the electrical load is part of an electroniccircuit operatively connecting the first porous anode and the secondporous cathode; wherein the first sodium conductive solid electrolytecomprises a β″-alumina electrolyte.
 21. The method of claim 20, whereinthe first sodium conductive solid electrolyte comprises a materialselected from the group consisting of: (Na₂O)_(1−x)11Al₂O₃ where x isfrom 0.1 to 0.75; Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; andNa_(1.67)(MgO)_(1.33)10.33Al₂O₃.
 22. The method claim 20, wherein thesecond sodium conductive solid electrolyte comprises a material selectedfrom the group consisting of: Na₃Zr₂Si₂PO₁₂; Na_(1+x)Si_(x)Zr₂P_(3−x)P₁₂where x is from 1.6 to 2.4; Y doped NaSICON of the formula(Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂, Na_(1+x)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12; Fe-doped NaSICON ofthe formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂, wherein Re isselected from the group consisting of Y, Gd, Dy, and Nd; andNa₄ZrSi₃O₁₂.
 23. The method of claim 20, wherein the second sodiumconductive solid electrolyte comprises a β″-alumina electrolyte.
 24. Amulti-stage sodium heat engine comprising: a first stage comprising afirst high pressure chamber which comprises a first sodium conductivesolid electrolyte, the first stage configured to provide expansion of afirst stream of vaporized sodium by ionizing the vaporized sodium toform sodium ions, the sodium ions being transferable through the firstsodium conductive solid electrolyte; a second stage comprising a secondlow pressure chamber which comprises a second sodium conductive solidelectrolyte, the second stage configured to provide expansion of asecond stream of vaporized sodium by ionizing the vaporized sodium toform sodium ions, the sodium ions being transferable through the secondsodium conductive solid electrolyte; and an electrical circuitoperatively connecting the first stage and the second stage with anelectrical load, wherein a Carnot efficiency of the multi-stage sodiumheat engine is at least 0.5; wherein the first sodium conductive solidelectrolyte comprises a β″-alumina electrolyte.
 25. The multi-stagesodium heat engine of claim 24, further comprising a third stagecomprising a third sodium conductive solid electrolyte, the third stageconfigured to provide expansion of a third stream of vaporized sodium byionizing the vaporized sodium to form sodium ions, the sodium ions beingtransferable through the third sodium conductive solid electrolyte,wherein the electrical circuit operatively connects the first stage, thesecond stage, and the third stage with the electrical load.
 26. Themulti-stage sodium heat engine of claim 24, wherein the second stagereceives the second stream of vaporized sodium from the first stage. 27.The multi-stage sodium heat engine of claim 24, wherein the first sodiumconductive solid electrolyte comprises a material selected from thegroup consisting of: (Na₂O)_(1−x)11Al₂O₃ where x is from 0.1 to 0.75;Na_(1.72)Al_(10.66)Li_(0.30)O₁₇; and Na_(1.67)(MgO)_(1.33)10.33Al₂O₃.28. The multi-stage sodium heat engine of claim 24, wherein the secondsodium conductive solid electrolyte comprises a material selected fromthe group consisting of: Na₃Zr₂Si₂PO₁₂; Na_(1+x)Si_(x)Zr₂P_(3−x)O₁₂where x is from 1.6 to 2.4; Y doped NaSICON of the formula(Na_(1+x+y)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O₁₂, Na_(1+x)Zr_(2−y)Y_(y)Si_(x)P_(3−x)O_(12−y)) where x is 2 and y is 0.12; and Fe-doped NaSICONof the formula (Na₃Zr_(2/3)Fe_(4/3)P₃O₁₂); Na₅ReSi₄O₁₂, wherein Re isselected from the group consisting of Y, Gd, Dy, and Nd; andNa₄ZrSi₃O₁₂.
 29. The multi-stage sodium heat engine of claim 24, whereinthe β″-alumina electrolyte comprises a sodium ion conductive solidelectrolyte of the formula Na₅LiAl₃₂O₅₁.
 30. The sodium heat engine ofclaim 1, wherein the first sodium conductive solid electrolyte isdifferent from the second sodium conductive solid electrolyte.
 31. Themulti-stage sodium heat engine of claim 24, wherein the first sodiumconductive solid electrolyte is different from the second sodiumconductive solid electrolyte.
 32. The multi-stage sodium heat engine ofclaim 24, further comprising a second heat source for providing aquantity of heat to the second stream of vaporized sodium between thefirst stage and the second stage.